The definite integral

The definite integral starts from a completely different idea than the indefinite one. Slice the region under a curve between a and b into thin rectangles, add their areas, then let the slices get thinner without bound. If those sums converge, the limit is the Riemann integral — a single number, not a function, standing for an accumulated total.

That framing is what makes it useful far beyond area. Slice a journey into short intervals where the velocity is nearly constant and the sum of v·Δt approaches the distance travelled. Slice a push into intervals where the force is nearly constant and the sum of F·Δx approaches the work done. Whenever a quantity is built by accumulating a rate over an interval, it is a definite integral.

The fundamental theorem of calculus is the surprise: this limit of sums can be computed by antidifferentiation. Find any F with F′ = f and the integral is F(b) − F(a). Two consequences follow directly. The +C is irrelevant here, because it cancels in the subtraction — which is why "any antiderivative" is enough. And the result is signed: stretches where the function is negative contribute negatively, so an integral can come out negative or zero even though the region plainly has area. If unsigned area is what you want, integrate the absolute value, or split the interval at the crossings and add the pieces.